Answer to Question #163678 in Statistics and Probability for Sanduni

Question

The annual number of earthquakes registering at least 2.5 on the Richter Scale and having an epicenter within 40 miles of downtown Memphis follows a Poisson distribution with mean 6.5.

a. What is the probability that at least 9 such earthquakes will strike next year?

b. What is the probability that at least 25 such earthquakes will strike during the next 5 years?

Answer 1

Solution:

a.

We need to find the probability of Poisson distribution wil $\lambda = 6.5$

Remembering Poisson formula and using calculator:

$P_n(\xi=m) = \frac{\lambda^m}{m!}e^{-\lambda}$

$P_n(\xi\ge9)=1- P_n(\xi=0)-P_n(\xi=1)-...-P_n(\xi=8)=$

$=1 - 0.0015 - 0.00977-0.03176-0.06881-0.11182-0.14537-$

$-0.15748-0.14623-0.11882\approx0.20843$

b.

We can use the other Poisson distribution, for 5 years, where $\lambda=6.5*5=32.5$

Using inverted probability:

$P_n(\xi\ge25)= 1- P_n(\xi \le 24)= 1 - P_n(\xi=0) - P_n(\xi=1) - ...-P_n(\xi=24) =$

$= 1- 0.07536 =0.92464$

Answer:

a. $P_n(\xi\ge 9)=0.20843$

b.$P_n(\xi\ge 25)=0.92464$